CAUCHY-RASSIAS STABILITY OF LINEAR MAPPINGS IN BANACH MODULES ASSOCIATED WITH A GENERALIZED JENSEN TYPE MAPPING

Abstract

Let X and Y be vector spaces. We show that a mapping f : X -> Y satisfies the functional equation, f (x(1) + Sigma(2d)(j=2)(-1)(j)x(j))-f (x(1) + Sigma(2d)(j=2)(-1)(j-1)x(j)) = 2 Sigma(2d)(j=2)(-1)(j) f(x(j)) j= 2 (- 1) j f( xj) if and only if the mapping f : X -> Y is Cauchy additive, and prove the Cauchy-Rassias stability of the above functional equation in Banach modules over a unital C*-algebra, and in Poisson Banach modules over a unital Poisson C*-algebra. Let A and B be unital C*-algebras, Poisson C*-algebras or Poisson JC*-algebras. As an application, we show that every almost homomorphism h : A -> B of A into B is a homomorphism when h(2(n) uy) = h(2(n)u) h(y) or h(2(n)u circle y) = h(2(n) u) circle h(y), for all unitaries u is an element of A, all y is an element of A, and n = 0, 1, 2, ... . Moreover, we prove the Cauchy-Rassias stability of homomorphisms in C*-algebras, Poisson C*-algebras or Poisson JC*-algebras.